First get a removable drive and back up all data you wish to keep, usually all C:\Users\ subfolders, so you don't lose any documents, pictures etc. you might want to retreive later.

Yes, all browsing and personalisation data is stored in Firefox profiles as files in your computer, and can be easily copied. If you could boot up, you could go to about:profiles and see which profile you are using and where it is stored. However, since you cannot run Firefox on the machine, try searching for the profile manually. Its default folder is C:\Users\[USERNAME]\AppData\Roaming\Mozilla\Firefox\Profiles\. When you open it, you will see some subfolders with names like $$$$$$$$.default where $s are random characters (so that folder names don't clash). Even the "default" part of the name can be different under certain circumstances. Profiles are created with each installation of Firefox, so multiple can exist on the same PC, even though only one is actively being used. If there are multiple of them and you don't know which one it is, copy all of them. The one you are looking for will probably have the largest size (since most sites were saved in history, bookmarks etc.) and newest date of modification. To copy a Firefox profile, simply copy the profile folder, probably named $$$$$$$$.default into the C:\Users\[USERNAME]\AppData\Roaming\Mozilla\Firefox\Profiles\ of another PC (or the same PC if you wiped it). Then, you need to "tell" Firefox to begin using the new profile on startup, which you can do using the Profile Manager. Feel free to search for help online or reply to ask, but most of your questions will probably be answered here: Recovering important data from an old profile Profile Manager - Create, remove or switch Firefox profiles

Two footnote:

(1) The AppData folder is hidden by default. To set Windows to show hidden files and folders, see:

https://support.microsoft.com/en-us/help/14201/windows-show-hidden-files

(2) Firefox has folders under both AppData/Roaming and AppData/Local. You only need to back up the one under Roaming. Unfortunately, we sometimes see users who backed up the one under Local rather than Roaming and there is no way to get bookmarks/logins back from the Local folder (it just has web cache and other temporary stuff).

(1) The AppData folder is hidden by default. To set Windows to show hidden files and folders, [...]

Also find such method for whatever "OS" your Live CD has but I suppose it should show hidden files by default as it is aimed at advanced users anyway.